The nonrealizability of modular rings of polynomial invariants by the cohomology of a topological space

Abstract:

Let$G < {\text {GL}}(n;{{\mathbf {F}}_p})$ be a $p$-group, $p$ an odd prime, and ${R^*}: = {{\mathbf {F}}_p}{[{x_1}, \ldots ,{x_n}]^G}$ the ring of invariants. The purpose of this note is to prove that in the case where ${R^*}$ is a graded polynomial algebra, where $\deg {x_1} = \cdots = \deg {x_n} = 2$, then there is no space $X$ such that ${H^*}(X:{{\mathbf {F}}_p}) \simeq {R^*}$. This complements the work of Clark and Ewing [3] and Adams and Wilkerson [1] on the case $p\nmid [G;1]$.